Robert Rubinstein

Nonlinear Reynolds stress models and the renormalization group

The renormalization group is applied to derive a nonlinear algebraic Reynolds stress model of anisotropic turbulence in which the Reynolds stresses are quadratic functions of the mean velocity gradients. The model results from a perturbation expansion that is truncated systematically at second order with subsequent terms contributing no further information. The resulting turbulence model applies to both low and high Reynolds number flows without requiring wall functions or ad hoc modifications of the equations. All constants are derived from the renormalization group procedure; no adjustable constants arise. The model permits inequality of the Reynolds normal stresses, a necessary condition for calculating turbulence‐driven secondary flows in noncircular ducts.

Effects of subgrid-scale modeling on time correlations in large eddy simulation

The effects of the unresolved subgrid-scale (SGS) motions on the energy balance of the resolved scales in large eddy simulation (LES) have been investigated actively because modeling the energy transfer between the resolved and unresolved scales is crucial to constructing accurate SGS models. But the subgrid scales not only modify the energy balance, they also contribute to temporal decorrelation of the resolved scales. The importance of this effect in applications including the predictability problem and the evaluation of sound radiation by turbulent flows motivates the present study of the effect of SGS modeling on turbulent time correlations. This paper compares the two-point, two-time Eulerian velocity correlation in isotropic homogeneous turbulence evaluated by direct numerical simulation (DNS) with the correlations evaluated by LES using a standard spectral eddy viscosity. It proves convenient to express the two-point correlations in terms of spatial Fourier decomposition of the velocity field. The LES fields are more coherent than the DNS fields: their time correlations decay more slowly at all resolved scales of motion and both their integral scales and microscales are larger than those of the DNS field. Filtering alone is not responsible for this effect: in the Fourier representation, the time correlations of the filtered DNS field are identical to those of the DNS field itself. The possibility of modeling the decorrelating effects of the unresolved scales of motion by including a random force in the model is briefly discussed. The results could have applications to the problem of computing sound sources in isotropic homogeneous turbulence by LES

The frequency spectrum of sound radiated by isotropic turbulence

Abstract The theory of turbulent time correlations is applied to compute the frequency spectrum of sound radiated by isotropic turbulence. Assuming the quasi-normality of the fourth-order velocity correlation generated by Lighthills theory, it is argued that the frequency distribution of radiated acoustic energy is determined by the Eulerian space-time correlation function. The frequency spectrum of the sound radiated by isotropic turbulence is found to scale as ω −4/3 at high frequencies. Dimensional analysis based on Kolmogorov scaling predicts the dependence ω −7/2 instead. The discrepancy is explained by the dependence of Eulerian time correlations on an additional dimensional parameter: the sweeping velocity of the most energetic scales of motion.

Anisotropic developments for homogeneous shear flows

The general decomposition of the spectral correlation tensor Rij(k) by Cambon et al. [J. Fluid Mech. 202, 295 (1989); 337, 303 (1997)] into directional and polarization components is applied to the representation of Rij(k) by spherically averaged quantities. The decomposition splits the deviatoric part Hij(k) of the spherical average of Rij(k) into directional and polarization components Hij(e)(k) and Hij(z)(k). A self-consistent representation of the spectral tensor is constructed in terms of these spherically averaged quantities. The directional and polarization components must be treated independently: representation of the spectral tensor using the spherical average Hij(k) alone proves to be inconsistent with Navier-Stokes dynamics. In particular, a spectral tensor consistent with a prescribed Reynolds stress is not unique. Since spherical averaging entails a loss of information, the description of an anisotropic correlation tensor by spherical averages is limited to weak departures from isotropy. The...

On consistency conditions for rotating turbulent flows

Consistency conditions for the prediction of turbulent flows in a rotating frame are examined. It is shown that the dissipation rate should vanish along with the eddy viscosity in the limit of rapid rotations. The latter result is also true when the eddy viscosity is anisotropic and formally follows from the explicit algebraic stress approximation as well as from a phenomenological treatment. The former result has been built into the modeled dissipation rate equation of recent turbulence models where the second result has been violated. In fact, some of these models have the eddy viscosity going to infinity while the dissipation rate vanishes, leading to an inconsistency. For consistency, both of these conditions must be satisfied. The implications of these results for turbulence modeling are thoroughly discussed.

Renormalization group analysis of anisotropic diffusion in turbulent shear flows

The renormalization group is applied to compute anisotropic corrections to the scalar eddy diffusivity representation of turbulent diffusion of a passive scalar. The corrections are linear in the mean velocity gradients. All model constants are computed theoretically. A form of the theory valid at arbitrary Reynolds number is derived. The theory applies only when convection of the velocity–scalar correlation can be neglected. A ratio of diffusivity components, found experimentally to have a nearly constant value in a variety of shear flows, is computed theoretically for flows in a certain state of equilibrium. The theoretical value is well within the fairly narrow range of experimentally observed values. Theoretical predictions of this diffusivity ratio are also compared with data from experiments and direct numerical simulations of homogeneous shear flows with constant velocity and scalar gradients.

Sweeping and straining effects in sound generation by high Reynolds number isotropic turbulence

The sound radiated by isotropic turbulence is computed using inertial range scaling expressions for the relevant two‐time and two‐point correlations. The result depends on whether the decay of Eulerian time correlations is dominated by large scale sweeping or by local straining: the straining hypothesis leads to an expression for total acoustic power given originally by Proudman, whereas the sweeping hypothesis leads to a more recent result due to Lilley.

Reynolds number effect on the velocity increment skewness in isotropic turbulence

Second and third order longitudinal structure functions and wavenumber spectra of isotropic turbulence are computed using the eddy-damped quasi-normal Markovian model (EDQNM) and compared to results of the multifractal formalism. It is shown that both the multifractal model and EDQNM give power-law corrections to the inertial range scaling of the velocity increment skewness. For the multifractal formalism, this is an intermittency correction that persists at any high Reynolds number. For EDQNM, this correction is a finite Reynolds number effect, and it is shown that very high Reynolds numbers are needed for this correction to become insignificant with respect to intermittency corrections. Furthermore, the two approaches yield realistic behavior of second and third order statistics of the velocity fluctuations in the dissipative and near-dissipative ranges. Similarities and differences are highlighted, in particular, the Reynolds number dependence.

A turbulence model for magnetohydrodynamic plasmas

The statistical theory of inhomogeneous turbulence is applied to develop a system of model equations for magnetohydrodynamic (MHD) turbulence. The statistical descriptors of MHD turbulence are taken to be the turbulent MHD energy, its dissipation rate, the turbulent cross helicity (velocity-magnetic field correlation), turbulent MHD residual energy (difference between the kinetic and magnetic energies), and turbulent residual helicity (difference between the kinetic and current helicities). Evolution equations for these statistical quantities are coupled to the mean-field dynamics. The model is applied to two MHD-plasma phenomena: turbulence evolution with prescribed mean velocity and magnetic fields in the solar wind, and mean flow generation in the presence of a mean magnetic field and cross helicity in tokamak plasmas. These applications support the validity of the turbulence model. In the presence of a mean magnetic field, turbulence dynamics should be subject to combined effects of nonlinearity and Alfvén waves; consequences for the dissipation rate of MHD residual energy are discussed.

Multiple-scale perturbation analysis of slowly evolving turbulence

A multiple-scale perturbation expansion is applied to extract a closed system of two equations governing the scalar descriptors of the turbulence energy spectrum from a spectral closure model. The result applies when the length scale and total energy input of a force that maintains a steady state of homogeneous isotropic turbulence are perturbed slowly and the energy spectrum consequently evolves slowly compared to the time scales of the turbulence itself.